
\magnification=1200
\nopagenumbers

\centerline{\bf Construction of the U(1) link element associated}
\centerline{\bf with the O(3) spin on sites}
\bigskip
Let $e_i$, $i=1,2,3$ denote the O(3) spin on each lattice site
with $e \cdot e =1$. Then we can write
$$\eqalign{
e_1 & =\sin\theta\cos\phi \cr
e_2 & =\sin\theta\sin\phi \cr
e_3 & =\cos\theta\cr }\eqno{(1)}$$
We assume that $\theta$ is in the range $[0, \pi)$. 

Next we define CP(1) variables on each site by
$$z_1 = \cos{\theta\over 2} e^{-i\phi/2};\ \ \ \ \
z_2 = \sin{\theta\over 2} e^{i\phi/2}
\eqno{(2)}$$

The U(1) variables on the links are then given by
$$U_\mu(n) = { z_1(n) z_1^*(n+\mu) + z_2(n) z_2^*(n+\mu) \over
|z_1(n) z_1^*(n+\mu) + z_2(n) z_2^*(n+\mu)| }
\eqno{(3)}
$$
\end

